Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$-6$$

Step-by-Step Solution

Answer

The modulus of \(-6\) is 6.

Step 1: Understand Modulus

The modulus (or absolute value) of a number is its distance from zero on the number line, without considering the direction. For any real number \(a\), the modulus \(|a|\) is always non-negative.

Step 2: Apply Modulus to the Given Number

Identify the given number, which is \(-6\). To find its modulus, we ignore the negative sign. Thus, the modulus of \(-6\) is \(6\).

Key Concepts

Absolute Value

Understanding absolute value is essential when dealing with real numbers. The absolute value of a number, represented by two vertical bars like \(|a|\), refers to the number's distance from zero on a number line.

- It measures magnitude without considering direction, meaning it is always non-negative. - For example, the absolute value of \(-6\) becomes \(6\) because \(-6\) is 6 units away from zero.

Remember, an absolute value strips away any negative sign. If you have a positive number or zero, its absolute value remains unchanged. In symbols: \[|a| = \begin{cases} a, & \text{if } a \geq 0 \-a, & \text{if } a < 0 \end{cases}\]

This property is helpful in many mathematical problems and real-world applications, as it provides a uniform way to discuss measures without the complication of direction.

Number Line

A number line is a straight representation of numbers in their natural order, extending infinitely in both directions. It's a helpful tool for visualizing real numbers and their relationships.

- The central point is zero, with positive numbers to the right and negative numbers to the left. - Every point on a number line is a real number and each number has a unique point, precisely one unit from its neighbors.

With a number line, you can easily locate the distance of a number from zero, such as when finding the absolute value. For instance, if you place \(-6\) on a number line, you physically measure 6 units from zero.

Visualizing numbers this way allows you to grasp the relationship between numbers better, perform operations like addition and subtraction more intuitively, and resolve problems involving distance and ordering with more confidence.

Real Numbers

Real numbers encompass most numbers you encounter, including whole numbers, decimals, and fractions. They can be classified as rational or irrational, positive, negative, or zero.

- **Rational numbers** include integers, fractions, and any number that can be expressed as a fraction of two integers. Examples are \(-3, 1/2, 0\). - **Irrational numbers** are numbers that cannot be written as fractions, like \(\pi\) or \(\sqrt{2}\).

Let’s consider real numbers within the context of absolute value: Whenever you find the absolute value of a real number, you are working with its modulus. This doesn't change the number's classification (rational or irrational) but focuses on its magnitude. Therefore, knowing the nature of real numbers helps you understand concepts like distance and magnitude on a broader spectrum.

Real numbers are important because they allow for comprehensive calculations and modeling of real-world scenarios. They help in understanding quantities that lie between integers, offering a complete mathematical description of any quantity.